Mathematical Analysis, 2nd edition – Tom M. Apostol

This is a classic fundamental text used in many universities in both undergraduate and graduate analysis. The book aids the reader in building a solid foundational understanding of exactly what it is that makes calculus work. This book was my constant companion throughout graduate school, and I took two courses out of it. The book is highly readable without being casual. The proofs are concise but well-motivated and thorough. (Not too many “proof is left to the reader” issues). Apostol’s writing helps the topics flow from one to another naturally. The topics build on one another, forming a constructive book that helps the reader finish the book with a solid structure of understanding of basic mathematical analysis. Numerous exercises are well-written and allow for good exploration in possible counterexamples of theorems, specific examples and illustrative applications. One other nice feature is that end of each chapter includes suggested references to complement the section itself, as opposed to a pile of references at the end of the text. It is possible to use this text for self-study in analysis, though I only recommend using this for self-study if one has a fairly solid background in calculus and logic first. Note: the international edition is also fine–identical to the original. 

— Rachel Traylor, Ph.D.



Univariate Calculus, some exposure to proofs and logic


Topics Covered

  • Real and complex number systems
  • Basic Set Theory
  • Point-Set Topology –  
    • Open and closed sets
    • Bolzano-Weierstrass Theorem
    • Cantor Intersection Theorem
    • Lindelof Covering Theorem
    • Compactness and metric spaces
  • Limits and Continuity
  • Derivatives 
  • Bounded Variation and Total Variation
  • Riemann-Stieltjes Integration
  • Residue Calculus
  • Infinite Series and Products
    • Convergence and tests for convergence
    • double sequences and double series
    • Cesaro summability
  • Sequences of Functions 
    • types of convergence 
    • power series
  • Lebesgue Integration
    • Levi monotone convergence theorems
  • Fourier series 
  • Multivariable Differentiable Calculus
  • Implicit Functions and Extrema
  • Multiple Integration: Riemann and Lebesgue



Difficulty 3
Proof Quality 5
Readability 4
Self-Study 4
Good for teaching? 5
Quality of Exercises 5